| L(s) = 1 | − 1.91·3-s − 1.51·5-s − 0.580·7-s + 0.682·9-s + 1.31·11-s + 3.89·13-s + 2.90·15-s − 1.20·17-s + 19-s + 1.11·21-s − 5.85·23-s − 2.70·25-s + 4.44·27-s + 1.29·29-s + 2.96·31-s − 2.52·33-s + 0.878·35-s + 1.18·37-s − 7.46·39-s + 9.04·41-s − 8.38·43-s − 1.03·45-s − 12.8·47-s − 6.66·49-s + 2.30·51-s + 3.07·53-s − 1.99·55-s + ⋯ |
| L(s) = 1 | − 1.10·3-s − 0.676·5-s − 0.219·7-s + 0.227·9-s + 0.397·11-s + 1.07·13-s + 0.749·15-s − 0.291·17-s + 0.229·19-s + 0.242·21-s − 1.22·23-s − 0.541·25-s + 0.855·27-s + 0.239·29-s + 0.532·31-s − 0.440·33-s + 0.148·35-s + 0.194·37-s − 1.19·39-s + 1.41·41-s − 1.27·43-s − 0.153·45-s − 1.87·47-s − 0.951·49-s + 0.322·51-s + 0.421·53-s − 0.268·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7620838961\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7620838961\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 1.91T + 3T^{2} \) |
| 5 | \( 1 + 1.51T + 5T^{2} \) |
| 7 | \( 1 + 0.580T + 7T^{2} \) |
| 11 | \( 1 - 1.31T + 11T^{2} \) |
| 13 | \( 1 - 3.89T + 13T^{2} \) |
| 17 | \( 1 + 1.20T + 17T^{2} \) |
| 23 | \( 1 + 5.85T + 23T^{2} \) |
| 29 | \( 1 - 1.29T + 29T^{2} \) |
| 31 | \( 1 - 2.96T + 31T^{2} \) |
| 37 | \( 1 - 1.18T + 37T^{2} \) |
| 41 | \( 1 - 9.04T + 41T^{2} \) |
| 43 | \( 1 + 8.38T + 43T^{2} \) |
| 47 | \( 1 + 12.8T + 47T^{2} \) |
| 53 | \( 1 - 3.07T + 53T^{2} \) |
| 59 | \( 1 + 0.258T + 59T^{2} \) |
| 61 | \( 1 - 14.7T + 61T^{2} \) |
| 67 | \( 1 + 9.54T + 67T^{2} \) |
| 71 | \( 1 - 6.93T + 71T^{2} \) |
| 73 | \( 1 + 15.2T + 73T^{2} \) |
| 79 | \( 1 + 13.1T + 79T^{2} \) |
| 83 | \( 1 + 3.70T + 83T^{2} \) |
| 89 | \( 1 - 8.36T + 89T^{2} \) |
| 97 | \( 1 - 17.0T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.260800117773492064291124300743, −7.53571566008181573158637879377, −6.51148977225057683718037257285, −6.23019718485462454715842018356, −5.45718231952205434364147248875, −4.54451598821779185295908311413, −3.89387472335136052750845261381, −3.04340968472366912486464813090, −1.66513895812740143376598575977, −0.51519546495614901978296837935,
0.51519546495614901978296837935, 1.66513895812740143376598575977, 3.04340968472366912486464813090, 3.89387472335136052750845261381, 4.54451598821779185295908311413, 5.45718231952205434364147248875, 6.23019718485462454715842018356, 6.51148977225057683718037257285, 7.53571566008181573158637879377, 8.260800117773492064291124300743
